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Statement

Theorem VI (p. 535). Let the weight $p(x)$ be non-negative and $L$-integrable in $[-1,1]$, and let the set of its roots (zeros) have measure $0$. Then, with suitable choice of the roots,

$$\lim_{n\to\infty}[\omega_n(z)]^{1/n}=\frac{z+\sqrt{z^2-1}}{2}$$

on the plane cut along $[-1,1]$, "uniformly in each interior domain" (p. 535, quoted).

Lemma VII (p. 534). Under the same hypotheses on $p$, for every $\epsilon>0$ and all sufficiently large $n$ (depending on $\epsilon$), the fundamental functions of the $p$-matrix satisfy $|l_{\nu,n}(x)|\le(1+\epsilon)^n$ for $-1\le x\le1$ and $\nu=1,\ldots,n$.

The introduction (p. 517) notes that the weight $e^{-1/x^2}$ satisfies the hypotheses while Szegő's asymptotic formula says nothing about it.

Proof pointer

P. 535: Lemma VII gives condition (41) of Theorem V, which gives the limit. Lemma VII (pp. 534--535) is proved by contradiction from Remez's inequality (an $n$th-degree polynomial bounded by $M$ on intervals of total length $\vartheta$ is bounded by $M|T_n(4/\vartheta-1)|$ on $[-1,1]$) and the Shohat minimum property of the Christoffel numbers (Lemma II, Corollary I).

Read depth

Claims checked: Theorem VI and Lemma VII were read clause by clause on the page images of the print; the proofs were followed for structure. Nothing here is independently reviewed.

Dependencies

Theorem V and Lemmas II and VII of the same paper; Remez's inequality.

Source. P. Erdős and P. Turán, On interpolation. III. Interpolatory theory of polynomials, Annals of Mathematics (2) 41 (3) (1940), 510--553, DOI 10.2307/1968733; the edition read is named on the source card.

Bears on

None of the problem pages directly.